Abstract
Steffensen-type methods with memory were originally designed to solve nonlinear equations without the use of additional functional evaluations per computing step. In this paper, a variant of Steffensen’s method is proposed which is derivative-free and with memory. In fact, using an acceleration technique via interpolation polynomials of appropriate degrees, the computational efficiency index of this scheme is improved. It is discussed that the new scheme is quite fast and has a high efficiency index. Finally, numerical investigations are brought forward to uphold the theoretical discussions.
Highlights
One of the commonly encountered topics in computational mathematics is to tackle solving a nonlinear algebraic equation
Before proceeding the given idea to improve the speed of convergence, efficiency index, and the attraction basins, we provide a short literature by reviewing some of the existing methods with accelerated convergence order
The motivation here is to know that is it possible to improve the performance of numerical methods in terms of the computational efficiency index, basins of attraction, and the rate of convergence without adding more sub-steps and propose a numerical method as a one-step solver
Summary
One of the commonly encountered topics in computational mathematics is to tackle solving a nonlinear algebraic equation. The motivation here is to know that is it possible to improve the performance of numerical methods in terms of the computational efficiency index, basins of attraction, and the rate of convergence without adding more sub-steps and propose a numerical method as a one-step solver. The aim of this paper is to design a one-step method with memory which is quite fast and has an improved efficiency index, based on the modification of the one-step method of Steffensen (Equation (1)) and increase the convergence order to 3.90057 without any additional functional evaluations. The rest of this paper is ordered as follows: In Section 2, we develop the one-point Steffensen-type iterative scheme (Equation (1)) with memory which was proposed by [18].
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